High-Frequency Asymptotic for the Forced Periodical Flows of an Inviscid Fluid through a Given Domain

Abstract

In this paper, we employ the method of multiple scales to study high-frequency time-periodic solutions for a boundary value problem for Euler equations in a domain with permeable boundary. Here the permeability means that an inflow and an outflow of fluid through some parts of the boundary are allowed. The boundary data given at the flow inlet and at the flow outlet are subjected to a high-frequency time-periodic modulation while the fluid is supposed to be free of mass forces. The relevant example is a flow through a finite duct or pipe affected by the modulation of the inflow and of the outflow of fluid. The keynote of our study is that the fluid passing through a domain is a non-conservative system even if the fluid is inviscid and incompressible. Indeed, the pumping and dissipation of energy occur when the fluid particles are entering the domain and, correspondingly, leaving it. In particular, this means that the flow should react on the time-periodic forcing like a non-conservative system (e.g. a viscous fluid). But since we deal with an inviscid fluid, the high-frequency modulation should produce short fast waves that propagate from the inlet downstream. They contribute into the leading term of asymptotic in spite of small amplitude they have in the case under consideration. Moreover, this contribution admits a simple explicit expression which makes the asymptotic and stability analysis for the time-periodic regimes readily accessible.